A213941
Partition array a(n,k) with the total number of bracelets (D_n symmetry) with n beads, each available in n colors, with color signature given by the k-th partition of n in Abramowitz-Stegun(A-St) order.
Original entry on oeis.org
1, 2, 1, 3, 6, 1, 4, 12, 12, 24, 3, 5, 20, 40, 60, 120, 120, 12, 6, 30, 90, 45, 180, 720, 220, 600, 1440, 900, 60, 7, 42, 126, 168, 315, 1890, 1050, 1890, 2100, 12600, 6720, 6300, 18900, 7560, 360, 8, 56, 224, 280, 224, 672, 4032, 6384, 5544, 6384, 5880, 45360
Offset: 1
n\k 1 2 3 4 5 6 7 8 9 10 11
1 1
2 2 1
3 3 6 1
4 4 12 12 24 3
5 5 20 40 60 120 120 12
6 6 30 90 45 180 720 220 600 1440 900 60
...
Row m=7 is: 7 42 126 168 315 1890 1050 1890 2100 12600 6720 6300 18900 7560 360.
For the rows n=1 to n=15 see the link.
a(3,1) = 3 because the 3 bracelets with 3 beads coming in 3 colors have the color multinomials (here monomials) c[1]^3=c[1]*c[1]*c[1], c[2]^3 and c[3]^3. The partition of 3 is [3], the color representative is c[1]^3, and the equivalence class with color signature from the partition [3] has the three given members. There is no difference between necklace and bracelet numbers in this case.
a(3,2) = 6 from the color signature 2,1 with the representative multinomial c[1]^2 c[2] with coefficient A213939(3,2) = 1, the only 3-bracelet cyclic(112) (taking j for the color c[j]), and A035206(3,2) = 6 members of the whole color equivalence class: cyclic(112), cyclic(113), cyclic(221), cyclic(223), cyclic(331) and cyclic(332). There is no difference between necklaces and bracelets numbers in this case.
a(3,3) = 1, color signature 1^3 = 1,1,1 with representative multinomial c[1]*c[2]*c[3] with coefficient A213939(3,3)=1 from the bracelet cyclic(1,2,3). The necklace (1,3,2) becomes equivalent to this one under D_3 operation. There are no other members in this class (A035206(3,3)=1).
The sum of row No. 3 is 10 = A081721(3). The bracelets are 111, 222, 333, 112, 113, 221, 223, 331, 332 and 123, all taken cyclically.
A214312
a(n) is the number of all four-color bracelets (necklaces with turning over allowed) with n beads and the four colors are from a repertoire of n distinct colors, for n >= 4.
Original entry on oeis.org
3, 120, 2040, 21420, 183330, 1320480, 8691480, 52727400, 303958710, 1674472800, 8928735816, 46280581620, 234611247780, 1166708558400, 5710351190400, 27565250985360, 131495088522060, 620771489730000, 2903870526350640, 13473567673441260, 62061657617625204, 283995655732351200
Offset: 4
a(5) = A213941(5,6) = 120 from the bracelet (with colors j for c[j], j=1, 2, ..., 5) 11234, 11243, 11324, 12134, 13124 and 14123, all six taken cyclically, each representing a class of order A035206(5,6) = 20 (if all 5 colors are used). For example, cyclic(11342) becomes equivalent to cyclic(11243) by turning over or reflection. The multiplicity 20 depends only on the color signature.
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t[n_, k_] := (For[t1 = 0; d = 1, d <= n, d++, If[Mod[n, d] == 0, t1 = t1 + EulerPhi[d]*k^(n/d)]]; If[EvenQ[n], (t1 + (n/2)*(1 + k)*k^(n/2))/(2*n), (t1 + n*k^((n + 1)/2))/(2*n)]);
a56344[n_, k_] := Sum[(-1)^i*Binomial[k, i]*t[n, k - i], {i, 0, k - 1}];
a[n_] := Binomial[n, 4]*a56344[n, 4];
Table[a[n], {n, 4, 25}] (* Jean-François Alcover, Jul 02 2018, after Andrew Howroyd *)
A214310
a(n) is the number of all three-color bracelets (necklaces with turning over allowed) with n beads and the three colors are from a repertoire of n distinct colors, for n >= 3.
Original entry on oeis.org
1, 24, 180, 1120, 5145, 23016, 91056, 357480, 1327095, 4893680, 17525508, 62254920, 217457695, 753332160, 2581110000, 8779264032, 29624681763, 99350001360, 331159123260, 1098168382080, 3624003213369, 11908069219816, 38972450763000, 127087400895000
Offset: 3
a(5) = A213941(5,4) + A213941(5,5) = 60 + 120 = 180 from the bracelet (with colors j for c[j], j=1, 2, ..., 5) 11123 and 11213, both taken cyclically, each representing a class of order A035206(5,4)= 30 (if all 5 colors are used), and 11223, 11232, 12123 and 12213, all taken cyclically, each representing a class of order A035206(5,5)= 30. For example, cyclic(11322) becomes equivalent to cyclic(11223) by turning over or reflection. The multiplicity A035206 depends only on the color signature.
A214313
a(n) is the number of all five-color bracelets (necklaces with turning over allowed) with n beads and the four colors are from a repertoire of n distinct colors, for n >= 5.
Original entry on oeis.org
12, 900, 25200, 442680, 5846400, 64420272, 622175400, 5466166200, 44611306740, 343916472900, 2531921456064, 17956666859040, 123458676825120, 827056125453600, 5419508203393200, 34847210197637424, 220424306985639540, 1374479672119161300, 8463477229726134000, 51536194734146965920, 310706598354410079360
Offset: 5
a(6) = A213941(6,10) = 900 from the bracelet with color signature [2,1,1,1,1] and color repertoire [c[j], j=1, 2, ..., 6]. There are A213939(6,10) = 30 bracelets with representative color multinomials c[1]^2 c[2] c[3] c[4] c[5]. If the colors c[j] are taken as j, e.g., 112345, 112354, 112435, 112453, 112534, 112543, 113245, 113254, 113425, (113452 is equivalent to 112543 by turning over), 113524, (113542 ==112453), 114235, ..., 121345, ... (all taken cyclically). Each of these 30 bracelets represents a class of A035206(6,10) = 30 bracelets when all six colors are used. Thus a(6) = 30*30 = 900 = 12*75.
A214308
a(n) is the number of all two colored bracelets (necklaces with turning over allowed) with n beads with the two colors from a repertoire of n distinct colors, for n>=2.
Original entry on oeis.org
1, 6, 24, 60, 165, 336, 784, 1584, 3420, 6820, 14652, 29484, 62335, 128310, 269760, 558960, 1175499, 2446668, 5131900, 10702020, 22385517, 46655224, 97344096, 202555800, 421478200, 875297124, 1816696728, 3764747868, 7795573230, 16121364000, 33310887808
Offset: 2
a(5) = A213941(5,2) + A213941(5,3) = 20 + 40 = 60 from the bracelet (with colors j for c[j], j=1,2,..,5) cyclic(11112) which represents a class of order A035206(5,2) = 20 (if all 5 colors are used), cyclic(11122) and cyclic(11212) each representing also a color class of 20 members each, summing to 60 bracelets with five beads and five colors available for the two color signatures [4,1] and [3,2].
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